Machine epsilon

Remarks

Example: in IEEE binary64, $\beta = 2$ and $p = 53$, so $\varepsilon_{\mathrm{mach}} = 2^{-52} \approx 2.22 \times 10^{-16}$ and $u = 2^{-53} \approx 1.11 \times 10^{-16}$. Check: the floating-point numbers in $[1, 2)$ are $1 + k \, 2^{-52}$ for $k = 0, 1, \ldots, 2^{52} - 1$, so the successor of $1$ is $1 + 2^{-52}$. Non-example: the smallest positive floating-point number, about $4.9 \times 10^{-324}$ in binary64, is not the machine epsilon; it measures the range, not the relative precision. If $x$ lies in the range of normalized numbers and $\mathrm{fl}(x)$ is the nearest floating-point number, then $\mathrm{fl}(x) = x(1 + \delta)$ with $\abs{\delta} \le u$.